A machine with a plurality of successive machining stations is known from U.S. Pat. No. 6,904,665. Tool turrets each having a grab are mounted to the left and right of each of two machining stations. The tool turrets are movable so that the machining and the gripping of the workpieces can take place in a machining station by tools and grabs of both tool turrets. Three tool is turrets are required for a machining apparatus operating according to this principle with two machining stations for “complete” machining, meaning not only machining outer surfaces of a rod-shaped workpiece centered on an axis by rotating the workpiece, lathe fashion, while pressing tools—cutting bits, grinding stones, burnishers—against its outer surface but also machining ends of the workpiece, for instance boring an axial hole in the workpiece end. Thus “complete” machining entails both engaging a tool radially with the workpiece while it is rotating about its axis and engaging a tool axially with the workpiece while rotating the tool and holding the workpiece against rotation or rotating the workpiece relative to a nonrotating tool.